(a) Field of the Invention
This invention relates to an electronic musical instrument and more particularly to an electronic musical instrument of a digital type which produces a tone signal by recursive calculation.
(B) Description of the Prior Art
Recently, such an electronic musical instrument has been developed that can produce not only the stationary musical tones, the tone pitch, the tone volume and the tone color of which are constant with respect to the lapse of time but also non-stationary musical tones, e.g., those of natural musical instruments, the tone pitch, the tone volume and the tone color of which vary in a complicated way with time. An electronic musical instrument provided with a recursive calculator was proposed for achieving generation of such complicated tones. (for example, U.S. Pat. application Ser. No. 689867)
Generally, a musical tone is considered to be a superposition of partial tones of particular frequencies (harmonics). Therefore, in the above type electronic musical instrument, a musical tone is divided into a plurality (M components) of partial tones f.sub.m (t) (m=1, 2, . . . ,M), the sample values f.sub.m (f.sub.m (nT) (n=0, 1, 2, . . . , and T is the sampling period) of the respective partial tones are calculated independently and recursively and the sum of the M partial tone sample values is calculated at each sampling time t = nT.
The recursive calculation for producing a partial tone function for constituting the partial tone of a musical tone is usually achieved in the following way. Namely, the recursive calculations perform calculations as: EQU f.sub.m (nT) = p.sub.m .multidot.f.sub.m [(n-1)T] + q.sub.m .multidot.f.sub.m [(n-2)T] (1)
where p.sub.m and q.sub.m are constants, or more generally ##EQU1## In formula (1), the sample value f.sub.m (nT) of the m-th partial tone at time t = nT is calculated from the preceding two sample values f.sub.m [(n-1)T] and f.sub.m [(n-2)T] of the same m-th partial tone at time t = (n-1)T and t = (n-2)T respectively, i.e., multiplying the respective recursion coefficients p.sub.m and q.sub.m and adding them together. Formula (2) represents a generalized form.
In such recursive calculation, the shapes of the partial tone functions can be varied widely depending on the selection of the parameters p.sub.m and q.sub.m and the initial values f.sub.m (0) and f.sub.m (T). In a preferred form, the recursive calculation is achieved in the following way.
For generating each of the partial tone functions EQU f.sub.m (t) = a.sub.m .multidot.exp(.alpha..sub.m t) .multidot.cos(.omega..sub.m t) (3)
according to the recursive calculation of formula (1), the two initial values are selected to be ##EQU2## and the coefficients are selected to be ##EQU3## Then, formula (1) leads to ##EQU4## Here, the relation ##EQU5## is used.
Such operations are achieved for the respective partial tones (practically in time-sharing manner) and added together to compose a sample value of the total musical tone F(t) at each sample time. Namely, ##EQU6##
As can be seen from the above description, the electronic musical instrument for generating musical tones by the above recursive calculation includes a recursive calculator for achieving the recursive calculations, recursion coefficient memory for storing M pairs of independent coefficients, p.sub.m (m=1 to M) and q.sub.m (m=1 to M), and initial value memory means for storing M pairs of independent initial function values, f.sub.m (0) (m=1 to M) and f.sub.m (T) (m=1 to M).
In such an electronic musical instrument, the amplitude a.sub.m of each partial tone f.sub.m (t) can be independently determined by the initial value f.sub.m (0) = a.sub.m, and the tone pitch and the envelope are determined by the other initial value and the two coefficients. Particularly, the envelope may be arranged constant by setting the coefficient .alpha..sub.m to be zero, attenuating by setting the coefficient .alpha..sub.m to be negative and diverging by setting the coefficient .alpha..sub.m to be positive. Thus, various modifications of the tone envelope can be performed according to this method. Therefore, it can be expected that such an electronic musical instrument can produce musical tones resembling close to the musical tones of the natural musical instruments.
Generally, the amplitudes of the respective partial tones of a musical tone differ from each other. Thus, when the number (M) of partial tones is large, the number (2M) of initial value memory elements for setting these partial tones also becomes very large.
Further, there exists further problem in synthesizing some kinds of musical tones, for example those of the piano, according to the above method. Namely, the amplitudes of the musical tones of the piano differ much in the higher pitch range and the lower pitch range and also they vary greatly by the depression strength of the key. Even considering piano tones of a particular pitch range, the amplitudes of the higher harmonics and the lower harmonics differ greatly. Thus, for insuring high fidelity production of piano tones, the initial value memories should store an extensively wide range of amplitudes, from the minimum to the maximum. This requires a large number of bits for each of the initial value memory units and the registers and the memories in the calculation circuit. Then, the scale of the calculator and hence the electronic musical instrument itself should inevitably becomes large.